C4 chapter 6 - integration

"In this chapter you will learn how to integrate more complicated functions."

C4 section 6.1

You need to be able to integrate standard functions.
Learn the standard integrals by playing with these integrals flashcards.












You need to know these standard integrals:

. . . . . .	1.	$\displaystyle \int{x^n} dx = \dfrac{x^{n+1}}{n+1} + C$
	2.	$\displaystyle \int {e^x} dx = e^x + C$
	3.	$\displaystyle \int{\dfrac{1}{x}} dx = \ln{|{x}|} + C$
	4.	$\displaystyle \int{\cos{x}} dx = \sin{x} + C$
	5.	$\displaystyle \int{\sin{x}} dx = -\cos{x} + C$
	6.	$\displaystyle \int{\sec^2{x}} dx = \tan{x} + C$
	7 .	$\displaystyle \int{\csc{x}\cot{x}} dx = -\csc{x} + C$
	8.	$\displaystyle\int{}csc^{2} {x} dx = -\cot{x} + C$
	9.	$\displaystyle \int{\sec{x}\tan{x}} dx = -\sec{x} + C$

You can practice offline with these:

Examples:

C4 Exercise 6A

• C4 Exercise 6A worked solutions
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  C4 Exercise 6A (standard integrals).jnt
  

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  C4 Exercise 6A (standard integrals).pdf
  

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C4 section 6.2

You should already know the first 9 standard integrals above and the chain rule for differentiation (see C3 chapter 8).
We are learning to: integrate functions which are identical to those in the standard results above except that x is replaced by the linear function ax + b.
General result: we can use the chain rule to find the correct constant to multiply by so that the derivative works out. In every case the new result is just like the integral before except x is replaced by ax + b and the whole result is multiplied by 1/a at the front.

You are strongly advised to learn these standard integrals off by heart. You could derive them by using the chain rule, but this will waste time in the exam:

. . . . . .	1.	$\displaystyle \int{(ax + b)^n} dx = \dfrac {1}{a} \dfrac{(ax + b)^{n+1}}{n+1} + C$
	2.	$\displaystyle \int {e^{ax + b}} dx = \dfrac {1}{a} e^{ax + b} + C$
	3.	$\displaystyle \int{\dfrac{1}{ax + b}} dx = \dfrac {1}{a} \ln{|{ax + b}|} + C$
	4.	$\displaystyle \int{\cos{(ax + b)}} dx = \dfrac {1}{a} \sin{(ax + b)} + C$
	5.	$\displaystyle \int{\sin{(ax + b)}} dx = - \dfrac {1}{a} \cos{(ax + b)} + C$
	6.	$\displaystyle \int{\sec^2{(ax + b)}} dx = \dfrac {1}{a} \tan{(ax + b)} + C$
	7 .	$\displaystyle \int{\csc{(ax + b)}\cot{(ax + b)}} dx = - \dfrac {1}{a} \csc{(ax + b)} + C$
	8.	$\displaystyle\int{}csc^{2} {(ax + b)} dx = - \dfrac {1}{a} \cot{(ax + b)} + C$
	9.	$\displaystyle \int{\sec{(ax + b)}\tan{(ax + b)}} dx = - \dfrac {1}{a} \sec{(ax + b)} + C$

Examples:

C4 Exercise 6B

• C4 Exercise 6B worked solutions:
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  C4 Exercise 6B (integration using reverse chain rule).jnt
  

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  C4 Exercise 6B (integration using reverse chain rule).pdf
  

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C4 section 6.3

You should already know the standard trig. identities from C3 chapter 6 (trigonometry) and C3 chapter 7 (further trigonometric identities and their applications).
We are learning to use our knowledge of standard trigonometric identities to simplify integrals
General result: replace a trig function you cannot integrate with one or more trig functions you can integrate
Here's the integration flow chart so far:

Standard identities to memorise (all these originally appear in C3 and earlier texts):

. . . . . .	$\tan {x} \equiv \dfrac {\sin{x}} {\cos{x}}$
	$\sin^{2} {x} + \cos^{2} {x} \equiv 1$
	$1 + \tan^{2} {x} \equiv \sec^2 {x}$
	$1 + \cot^{2} {x} \equiv \csc^{2} {x}$
	$\cos{2x} \equiv \cos^{2} {x} - \sin^{2} {x}$ $\implies \sin^{2} {x} \equiv \frac{1}{2} - \frac{1}{2}\cos{2x}$ $\implies \cos^{2} {x} \equiv \frac{1}{2} + \frac{1}{2}\cos{2x}$

Examples:

C4 Exercise 6C

• C4 Exercise 6C worked solutions:
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  C4 Exercise 6C (integration using trig identities) reloaded.jnt
  

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  C4 Exercise 6C (integration using trig identities) reloaded.pdf
  

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C4 section 6.4

You should already know how to write polynomial fractions using partial fractions from C4 chapter 1
We are learning to use our knowledge of partial fractions to simplify integrals.
General result: replace a polynomial fraction you cannot integrate with partial fractions you can integrate


Examples:

C4 Exercise 6D

• C4 Exercise 6D worked solutions:
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  C4 Exercise 6D (integration with partial fractions).jnt
  

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  C4 Exercise 6D (integration with partial fractions).pdf
  

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C4 section 6.5

You should already know the standard integrals and other techniques from earlier in C4 chapter 6, in particular the results of sections 6.1 and 6.2
We are learning to use our knowledge of derivatives to spot two common patterns to integrals:

1. a derivative of a function multiplied by a power of that function and
2. a derivative of a function as the numerator of a fraction over the function itself as the denominator of the function.

General result: spot integrals of the form:
$\displaystyle \int k \dfrac{\text{f'}(x)}{\text{f}(x)} \text{d}x \text{ try } \ln{|\text{f}(x)|} \text{ and adjust the constant.}$

or

$\displaystyle \int k \text{f'}(x) [\text{f}(x)]^n \text{d}x \text{ try } [\text{f}(x)]^{n+1} \text{ and adjust the constant.}$

Examples:

C4 Exercise 6E

• C4 Exercise 6E worked solutions:
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  C4 Exercise 6E (standard patterns in integration).jnt
  

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  C4 Exercise 6E (standard patterns in integration).pdf
  

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C4 section 6.6

You need to know how to carry out implicit differentiation from C4 section 4.2 and you need to be familiar with the range of techniques we have already met in C4 chapter 6, particularly the standard results and the work on partial fractions

We are learning to make a substitution into an integral defining u = f(x) and replacing the "dx" part of the integral with something appropriate involving "du." By this process, when we choose the function for u carefully, the result of this substitution is to make the integral substantially easier to integrate. In most of the examples and questions you will meet, the art of choosing the correct substitution is done for you, but with practice you should be able to predict what suitable substitution might work.

C4 Exercise 6F

• C4 Exercise 6F worked solutions (only partially complete - ask on the sandbox if you need more):
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  C4 Exercise 6F (integration by substitution).jnt
  

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  C4 Exercise 6F (integration by substitution) (partially complete).pdf
  

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C4 section 6.7

You need to know the product rule for differentiation from C3 section ## and you need to be familiar with the range of techniques we have already met in C4 chapter 6, particularly the standard results.

We are learning to exchange one integral for another by using a technique known as integration by parts. This technique arises from integrating a rearrangement of the product rule. It allows us to spot occasions where, after rejecting* all the simpler techniques above, we notice that the integral we seek is a product of something we can differentiate with something we can integrate although we cannot integrate the whole product directly. We can then rewrite the integral in an alternative, and hopefully better, form.

*Note: there's something very appealing to students about integration by parts and it is a common error to attempt to use it before checking that the integral doesn't succumb to a better method from sections 6.1 - 6.6. Please get in the habit of checking for those first and only use integration by parts when the others have failed.

Here's the formula:
$\displaystyle \int u \dfrac {\text{d}v}{\text{d}x} \text{d}x \equiv uv - \int v \dfrac{\text{d}u}{\text{d}x} \text{d}x$

This section also introduces four new standard integrals:

. . . . . .	$\displaystyle \int \tan{x} \text{d}x \equiv \ln{|\sec{x}|} + C$
	$\displaystyle \int \sec{x} \text{d}x \equiv \ln{|\sec{x} + \tan{x}|} + C$
	$\displaystyle \int \cot{x} \text{d}x \equiv \ln{|\sin{x}|} + C$
	$\displaystyle \int \csc{x} \text{d}x \equiv -\ln{|\csc{x} + \cot{x}|} + C$

You'll need to learn these off-by-heart along with the eighteen introduced earlier.
It's not introduced as a standard result, so you need to learn the rather unusual proof, but it's probably worth learning that:

. . . . . .

$\displaystyle \int \ln{x} \text{d}x \equiv x\ln{x} - x + C$

Here's the proof, using integration by parts.
Notice that the light-bulb moment is to write ln x as (ln x)×1
##Insert a screenshot of the proof written in Windows Journal##

C4 Exercise 6G

• C4 Exercise 6G worked solutions (more to follow):
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  C4 Exercise 6G (integration by parts).jnt
  

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• pdf to follow (ask on the sandbox if you need it urgently)